Question

Claudia has $12$ coins, each of which is a $5$ -cent coin or a $10$ -cent coin. There are exactly $17$ different values that can be obtained as combinations of one or more of her coins. How many $10$ -cent coins does Claudia have?

$3$

$4$

$5$

$6$

$7$

Answer: C

Concepts:money counting integers in a range

Difficulty rating: 1480

Solution:

Let the number of $5$ -cent coins be $x$ and the number of $10$ -cent coins be $12 - x.$

If $x=0,$ the only possible values are $10,20,\ldots,120$ cents, giving just $12$ values. Hence $x>0.$ Having at least one $5$ -cent coin then makes every multiple of $5$ from $5$ through the total value $5x+10(12-x)=120-5x$ obtainable.

There are $24 - x$ such multiples of $5,$ which means that $x = 7$ to get $17$ possible different values.

The number of $10$ -cent coins is therefore $12 - 7 = 5.$

Thus, C is the correct answer.

2015 AMC 10A Problem 13

Answer: C

Difficulty rating: 1480

Solution:

Problem 13 in Other Years